Power Reduction Formulas/Cosine to 4th/Proof 2

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Theorem

$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$


Proof

\(\displaystyle \cos 4 x\) \(=\) \(\displaystyle \paren {\frac {e^{i x} + e^{-i x} } 2}^4\) Cosine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i x} + e^{-i x} }^4} {16}\) rearranging
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i x} }^4 + 4 \paren {e^{i x} }^3 \paren {e^{-i x} } + 6 \paren {e^{i x} }^2 \paren {e^{-i x} }^2 + 4 \paren {e^{i x} } \paren {e^{-i x} }^3 + \paren {e^{-i x} }^4} {16}\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{4 i x} + 4 e^{2 i x} + 6 + 4 e^{-2 i x} + e^{-4 i x} } {16}\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 + 4 \paren {\dfrac {e^{2 i x} + e^{-2 i x} } 2} + \paren {\dfrac {e^{4 i x} + e^{-4 i x} } 2} } 8\) gathering terms
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 + 4 \cos 2 x + \cos 4 x} 8\) Cosine Exponential Formulation

$\blacksquare$


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